Abstract

A general fast numerical algorithm for chirp transforms is developed by using two fast Fourier transforms and employing an analytical kernel. This new algorithm unifies the calculations of arbitrary real-order fractional Fourier transforms and Fresnel diffraction. Its computational complexity is better than a fast convolution method using Fourier transforms. Furthermore, one can freely choose the sampling resolutions in both x and u space and zoom in on any portion of the data of interest. Computational results are compared with analytical ones. The errors are essentially limited by the accuracy of the fast Fourier transforms and are higher than the order 10-12 for most cases. As an example of its application to scalar diffraction, this algorithm can be used to calculate near-field patterns directly behind the aperture, 0⩽z<d2/λ. It compensates another algorithm for Fresnel diffraction that is limited to z>d2/λN [J. Opt. Soc. Am. A 15, 2111 (1998)]. Experimental results from waveguide-output microcoupler diffraction are in good agreement with the calculations.

For κ=2n the kernel becomes a Dirac delta function, BFrFT(2n)(u, x)=δ[u-(-1)nx], and the transform is straightforward and needs no further calculation.

To clarify the later results and be self-consistent, we will adopt the following definition of Fourier transform in the discrete form: Given f(x), its Fourier transform is g(u)≡F{f(x)}=∫-∞∞f(x)exp(-2πixu)dx, which could be numerically approximated by g(u)=F{f(x)}≈gk=∑l=0Nx-1fn exp[-2πi(n-Nx/2)(k-Nu/2)δxδu]δx. We have assumed that the Fourier transform will map f(x) from x∈[-(Nxδx)/2,+(Nxδx)/2] to g(u) in the domain u∈[-(Nuδu)/2,+(Nuδu)/2]. If gk is given by a standard DFT or FFT, however, the mapped domain will be u∈[-1/(2δx),+1/(2δx)] owing to the sampling condition δxδu≡1/Nx.

The width of the mask used to fabricate the 10-µm-thick waveguide is 50 µm. Owing to the highly isotropic etching, the final width of the polyimide waveguide can be varied in the range of 60∼70 µm depending on the precise control of experiment environments. In our simulation, we used a typical value of 65 µm.

The width of the mask used to fabricate the 10-µm-thick waveguide is 50 µm. Owing to the highly isotropic etching, the final width of the polyimide waveguide can be varied in the range of 60∼70 µm depending on the precise control of experiment environments. In our simulation, we used a typical value of 65 µm.

For κ=2n the kernel becomes a Dirac delta function, BFrFT(2n)(u, x)=δ[u-(-1)nx], and the transform is straightforward and needs no further calculation.

To clarify the later results and be self-consistent, we will adopt the following definition of Fourier transform in the discrete form: Given f(x), its Fourier transform is g(u)≡F{f(x)}=∫-∞∞f(x)exp(-2πixu)dx, which could be numerically approximated by g(u)=F{f(x)}≈gk=∑l=0Nx-1fn exp[-2πi(n-Nx/2)(k-Nu/2)δxδu]δx. We have assumed that the Fourier transform will map f(x) from x∈[-(Nxδx)/2,+(Nxδx)/2] to g(u) in the domain u∈[-(Nuδu)/2,+(Nuδu)/2]. If gk is given by a standard DFT or FFT, however, the mapped domain will be u∈[-1/(2δx),+1/(2δx)] owing to the sampling condition δxδu≡1/Nx.

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